Prototype Robot Parts
Some basic craft supplies are all you need to start playing around with the mechanical side of robotics. We're going to focus mainly on arms and hands, but you could apply the same techniques to other parts. We're also going to take some inspiration from biology; so while you're working have a think about how the principles apply to your own body.
Most of the projects covered were compiled on the puppets2robots site. That page also provides a lesson plan so that you can use the materials in a classroom setting.
Basic Arm
You can build a basic "arm" with just cardboard, string, drinking straws, a pivot (brass fastener), and tape. The original instructions come from the PBS Kids site, and the assembly is shown in the diagram below:

Pulling on the string makes the arm raise; letting out the string allows the arm to fall. In this set-up it is gravity that pulls the arm back down, so it only works in one orientation. In your own body most of the muscles work in pairs (known as antagonistic pairs) that pull in opposite directions. Using this principle you can easily add a second string to your robot arm so that it works in any direction, as shown below:
In a biological sense the string would be a muscle - which is fixed at both ends and contracts. In a robot this could be pneumatic or hydraulic pistons; or even just a string wound up by a motor.
Straw Fingers
Not every joint has its own pair of muscles. A prime example is your hands. As you move your fingers you'll notice muscles moving in your forearm. It's these muscles that pull on tendons (nature's strings) that make your fingers move.
You can easily make an equivalent yourself using our old friends drinking straws and string. Again this is covered by the puppets2robots site (with one example taken from this instructable article), but essentially you fix the string at one end of the straw and cut notches for the flex points.
There are lots of options for what you can do with this. The example below shows an arm extension, similar to what you might find in some puppets. The fingers are quite simple because the direction that you move your finger in matches the direction that you want the straw to bend in. The thumb is slightly trickier, but you can get a reasonable result by feeding the string around the back.
Gripper
A robot doesn't always need the complexity of a full hand though. For many tasks a simple gripper may suffice - similar to just your thumb and forefinger. However, even with something this simple there are still options and choices to be made - such as whether it should normally (i.e. when there is no power being applied) be open or closed. A design that lets you explore some of these options is shown below. You can use cardboard, but foam board works better.
You'll need a pivot point for them to rotate around; and a nail pushed through the pieces and hammered into a bit of wood works perfectly well. After that you can begin exploring different configurations. I got a bit fancier and connected metal hoops on the edges, but you could just tape the string on.
The video below shows some examples that you can try. Somehow it ended up looking like a late 1980s Open University science video, but it should be clear enough. In each case there is an image that shows the forces on a single piece. If you're building it yourself be sure to mirror the connections - e.g. if the string connects to the left of the top piece, then it should connect to the right of the bottom piece.
Universal Jamming Gripper
Although most of the prototypes so far have taken some inspiration from biology, we're not limited to just copying things from nature. A great example of something original is the universal gripper developed by the Cornell Computational Synthesis Laboratory, using coffee and a party balloon.
There are quite a few guides on how to create your own, but this instructable guide from Jason Poel Smith is one of the best. I had some reasonable success in making my own. I used instant coffee granules instead of ground coffee, because that's what I had easy access to, and although I tried using an air pump (designed to work with air beds), I found that my lungs gave better results. You need to inflate the balloon quite a bit before gripping to get good coverage, and maintain the vacuum (suction) in order to hold objects.